Continuous
Limits and Intro to Derivatives
Bren Calculus Workshop
Nathaniel Grimes
Bren School of Environmental Science
Last updated: Sep 15, 2024
Team Review
How did everyone feel about the problem set?
Any questions?
Discuss with Team
Disclaimer
Functions are like baking receipes
Assemble all your ingredients (independent variables like \(x_1\), \(x_2\))
Follow the instructions to mix, bake, and decorate (Instructions of function)
End up with final product (\(f(x)\))
Typically we use the notation \(f(x)=x\), but we can always use different representations like \(g(x)=x\) or \(y=x\)
Properties of functions
For each combination of independent variables, there is exactly one value of the dependent variable.
\(f(1)\ne\{2,3\}\) i.e. if I put 1 into the function I can’t have both 2 and 3 come out
Vertical line test
Functions can be continuous or discontinuous
Continuous: For every value of x, f(x) returns a number
Discontinuous: Parts are undefined
Are these functions
Continuous
Discontinuous
Limits
Definition:
The value a function approaches as the input approaches a specific value
\[ \large \lim_{x\to c} f(x)=L \]
Verbally:
“The limit of the function f(x) as x approaches the value c is L”
Start with an example
Strategy to solve: Evaluate f(x) at x getting closer and closer to 2
Quick sanity check, what should \(f(2)=?\)
\(f(2)=4\)
From both directions it looks like f(x) converges to 4
\[ \lim_{x\to c}(2x^2-4)=4 \]
Graphical Solution
Finding limits of functions when f(x) is continuous at c is easy.
\[ \lim_{x\to c}f(x)=f(c) \]
What happens with discontinuous functions?
Try with your group the same approach for:
\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]
Hint: Start very close to 2 like 1.9 and 2.1
We broke the universe
Dividing by zero is impossible
The function approaches different values from either side
Therefore…
Clearer on a graph
Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)
Defined Limit at an undefined point
Discontinuous functions can still have limits
Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2
x=2 may not exist, but we can still find a limit because it consistently approaches 3 from both directions
Work with your team to complete these task
Part 1:
\[ \lim_{x\to3}g(x)=0 \]
\[ \lim_{x\to-4}g(x)=3 \]
\[ \lim_{x\to 3} g(x)= \text{DNE} \]
Work with your team to complete these task
Part 2:
Can you think of examples where discontinuous functions might exist in environmental science?
Choose a a team to draw an example of one of these statements:
\(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite
\(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)
\(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)
\(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)
Putting it all together
Recall average rate of change and instantaneous
Taking the average rate of change to a set limit will eventually converge to the instantaneous.
Walk Trhough Example
Walk Through Example
Walk Through Example
Walk Through Example
Walk Through Example
Walkthrough
Tangent Lines
What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).
These are called tangent lines
Put our example in math notation
\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]
Choose a point along the function \((x,f(x))\)
Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)
Add these into the slope equation
\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]
Derivative Definition
\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]
Most common notation
\[ \large f'(x)\text{, or }\frac{dy}{dx} \]
Not all functions are differentiable
All differentiable functions are continuous, but not all continuous functions are differentiable
The absolute value function is one example \(y=|x|\)
Calculus and Derivatives are the study of change
Environmental Science is also a study of change
Environmental Science is also a study of change
Environmental Science is also a study of change
Rules for Differentiation
Constant Rule
\[
\begin{align}
&y=a &\frac{dy}{dx}=0
\end{align}
\]
Power Rule
\[ \frac{d}{dx}[x^n] =nx^{n-1} \]
Examples
\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]
Rules for Differentiation
Sum and Difference Rules
\[
\begin{align}
\frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\
\frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x)
\end{align}
\]
Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.
Example
\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]
As a team, list 5 fields of environmental science where studying the rate of change and derivatives would be important
Which rules should you use to take these derivatives?
\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]
\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]